This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I want to talk about the composition of
functions. Let’s say you’re given two functions.
You’ve got function f, and this is f(x)=x + 3, and function g, and that’s g(x)=x^2 – 5. and then
you’re told to find this thing — inside parentheses you’ve got an f, a small circle
and a g and then you close the parentheses,
open up another one and put an x in there. This is called the
composition of these two functions. We read this as “f compose g of x.” Now another way to write this which is much more convenient,
would be f, and after the f you want parenthesis, of g, and after g you want parenthesis, of x, and then you close those parentheses.
So this is f of g of x, and all we’re going to do is we’re going to work our way through this
by starting in the innermost parentheses.
We’re gonna find what g(x) is. So we’re told what g(x) is. g(x) is x^2 – 5. So I’ll take
g(x) and replace it with x^2 – 5. So that means
I’ll have f(x^2 -5). Now I have to find out what this is. Well I know what f(x) is. x + 3, and if I change
what’s inside those parentheses, instead of
an x I’ll have an x^2 – 5, what that means is I’ll
take the x + 3 and wherever I have an x
I’ll replace it with x^2 – 5. So f(x^2 – 5) is going to be….
I’ve got this x here, I’m replacing that with x^2 -5 and then + 3. Then I’ll just evaluate
what this is. I don’t need the parentheses,
so I can turn this into x^2 – 5 + 3. And that just becomes x^2 – 2. Okay, so f compose g of x turns into f(g(x)) — that’s just rewriting your
original problem. Then we plug in what g(x) is,
which we’ve been given. And then we find the f of that x^2 – 5, f(what g(x) was). Now if we
rearrange these, if we try to find out what GE g compose f of x is we’re very likely going to get a
different answer. In other words this is not like multiplication
where it doesn’t matter what order you do things in. So here we have g compose f of x.
I’m gonna turn that into g(f(x)). I want to start by find f(x), which I know. f(x) is x + 3.
So I’m going to replace f(x) with x + 3.
That means I have g(x + 3). My g(x) function tells me to take whatever I have inside these parentheses
here and square it and subtract 5. So I’m gong to take this x + 3 and square it and subtract 5. So let’s square x + 3.
That’s like x + 3 times x + 3. If we FOIL that out,
we’re going to get x^2 + 6x + 9 minus 5. And then
we’ll combine this 9 and the -5, so I’ll have x^2 + 6x…
9 – 5 is positive 4. So this is a very
different answer than what I got for the f(g(x)). So, going through this
one more time, what you do…
I’ll do the g compose f of x… what we do is we rewrite this as a function of a function, g(f(x)). Then we know what that innermost function is,
the f(x), because we’re given that. So we take whatever f(x) is —
in this case is was x + 3 — and we put that where
the f(x) was. So now we have g(x+ 3).
Now all we do is take that g(x) function and replace the x with x + 3, which gives us in this
case (x + 3), that whole expression
squared, minus 5. And then we simplify that
and we get our answer. Okay. So I hope this helps. Take care.
I’ll see you next time.